Pathway Realisability in Chemical Networks

TL;DR

A method for analysing the realisability of pathways based on the reachability question in Petri nets is presented and two extended notions of realisability of pathways are presented, one of which is related to the concept of network catalysts.

Abstract

The exploration of pathways and alternative pathways that have a specific function is of interest in numerous chemical contexts. A framework for specifying and searching for pathways has previously been developed, but a focus on which of the many pathway solutions are realisable, or can be made realisable, is missing. Realisable here means that there actually exists some sequencing of the reactions of the pathway that will execute the pathway. We present a method for analysing the realisability of pathways based on the reachability question in Petri nets. For realisable pathways, our method also provides a certificate encoding an order of the reactions which realises the pathway. We present two extended notions of realisability of pathways, one of which is related to the concept of network catalysts. We exemplify our findings on the pentose phosphate pathway. Furthermore, we discuss the relevance of our concepts for elucidating the choices often implicitly made when depicting pathways. Lastly, we lay the foundation for the mathematical theory of realisability.

Figures (25)

• Figure 1: A directed hypergraph in (a) and the corresponding bipartite graph in (b).
• Figure 2: Example of an extended hypergraph. It has vertices $\{A,B,C,D\}$, edges $\{e_1,e_2,e_3,e_4\}$, and a half-edge to and from each vertex. An edge $e$ is represented by a box with arrows to (from) each element in $e^-$ ($e^+$).
• Figure 3: Example flow $f$ on the extended hypergraph from Fig. \ref{['fig:hypergraph']}. Vertex $D$ has been omitted as it has no in- or out-flow. Edges leaving or entering $D$ have also been omitted as they have no flow. The flow on an edge is represented by an integer. For example, the half edge into $B$ has flow $f(e_B^-)=2$, the half edge leaving $B$ has flow $f(e_B^+)=1$, and edge $e_1$ has flow $f(e_1)=2$.
• Figure 4: Example firing sequence. Here $P=\{p_1,p_2,p_3,p_4,p_5\}$, $T=\{t_1,t_2,t_3\}$, $W=\{(p_1,t_1)\mapsto 1,(p_2,t_1) \mapsto 1, (t_1,p_3) \mapsto 1,(p_3,t_2)\mapsto 1,(t_2,p_4) \mapsto 1,(p_4,t_3) \mapsto 1,(t_3,p_5) \mapsto 1,(t_3,p_1) \mapsto 1\}$, and the initial marking $M_0=\{p_1\mapsto1, p_2\mapsto 1, p_3\mapsto 0, p_4\mapsto 0, p_5\mapsto 0\}$ which is depicted in (a). The firing sequence that leads to (d) is $\sigma=t_1t_2t_3$, which is illustrated through (a) to (d).
• Figure 5: The flow from Fig. \ref{['fig:hyperflow']} converted to a Petri net with its initial marking. Places are circles, transitions are rectangles, and tokens are black dots. Arrows indicate pairs of places and transitions for which the weight function $W$ is non-zero (in this example, all non-zero weights are equal to one). The target marking is $M_T(A_T)=1$, $M_T(B_T)=1$, $M_T(C_T)=1$ and $M_T(p)=0$ for all $p\in P \setminus \{A_T, B_T, C_T\}$.We have omitted the part of the net that corresponds to the omitted part of Fig. \ref{['fig:hyperflow']}.

Theorems & Definitions (20)

• Definition 3.1
• Definition 3.2: Occurrence Net goltz:83
• Definition 3.3: Process goltz:83 (adapted)
• Definition 3.4
• Definition 4.1: Scaled-Realisable
• Definition 4.2: Borrow-Realisable
• Definition 6.1: Flow-induced Subhypergraph
• Definition 6.2: König Representation Andersen:20
• Lemma 1
• proof